The action, by which the vertices of the multipole moments in the diagrams Ref. [1]} are defined has been
derived using group–theoretical methods in \(D=4\)  dimensions in Ref. [2]}, Eq. (100), e.g., and reads
\(S_{\rm mp} &=& - \sqrt{32 \pi G_N}\Biggl \lbrace \int _{-\infty }^{+\infty } dt [E h_{00} + L^i \varepsilon _{ijk} \partial _j h_{0k}+ \sum _{l=2}^\infty \frac{1}{l!} I^L \partial _{L-2} E_{k_l-1, k_l}\nonumber \\ && \hspace*{56.9055pt}- \sum _{l=2}^\infty \frac{2l}{(l+1)!} J^L \partial _{L-2} B_{k_l-1, k_l} + ...\Biggr \rbrace ,\) 
